Intro to Integration
Bren Calculus Workshop
Nathaniel Grimes
Bren School of Environmental Science & Management
Last updated: Sep 16, 2024
Team Review
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Any questions?
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But first we’ll start with natural logarithms
By Definition
\[ \large e^{\ln(x)}=x \]
Properties
\(\ln (e^x)=x\)
\(\ln(xy)=\ln x+\ln y\)
\(\ln(\frac{x}{y})=\ln x-\ln y\)
\(\ln(x^y)=y \ln x\)
Derivative of Natural Log
\[ \large \frac{d}{dx}[\ln x]=\frac{1}{x} \]
Math uses:
Allows us to more easily handle exponents \((e^x)\)
Surprisingly simple derivative
Really important in Integration
Real World Uses
Time and Growth Problems
Differential Equations
Examples of Natural Logs
Solve for t in this equation:
\[ A=Pe^{rt} \]
\[ \begin{align} \frac{A}{P}&=e^{rt} \\ \ln(\frac{A}{P})&=rt \\ \frac{\ln(A)-\ln(P)}{r}&=t \end{align} \]
Find the Derivative
\[ \large y=\ln(x^2) \]
\[ \begin{align} y=\ln(u)& &u=x^2\\ \frac{dy}{du}&=\frac{1}{u} &\frac{du}{dx}=2x \\ \frac{dy}{du}\frac{du}{dx}&=\frac{2x}{u} &\text{By Chain Rule}\\ \frac{dy}{dx}&=\frac{2x}{x^2}\\ \frac{dy}{dx}&=\frac{2}{x}\\ \end{align} \]
\[ \begin{align} \text{Differentiate: A) }f(x)=x \ln x& &\text{B) }5+\ln(3x)=7 & &\text{C) } e^{5x-0.2}=10 \end{align} \]
The increase in your funds is given by: \[S=122000e^{0.032 t}\]
where S is the amount in your retirement account t years after leaving the company.
How many years after you quit will your retirement account be earning $8000/year in interest?
How do we find the area under this curve?
We can approximate the area by summing the area of easy to calculate rectangles
How do we make the rectangles
Start at one end of the interval \([a,b]\)
Evaluate the function at \(f(a)\)
Step away from \(a\) by some small amount called \(\Delta x\)
Multiply \(f(a)\Delta x\)
Repeat the same steps above, but keep moving the function evaluation to new intervals
Sum up all (k) rectangles
\[ \text{Area}=f(x_1)\Delta x+f(x_2)\Delta x+...f(x_k)\Delta x \]
This is called a Reimann Sum
Example of Reimann Sum
Approximate the area under the curve from \([-3,3]\) with \(\Delta x=1\) and \(k=6\)
| x | y |
|---|---|
| -3 | 11 |
| -2 | 18 |
| -1 | 13 |
| 0 | 2 |
| 1 | -9 |
| 2 | -14 |
| 3 | -7 |
Expanding Reimann Sums
What if we try to make \(\Delta x\) really small and the number of rectangles really big \((k\to \infty)\)?
Our estimation of the Area will get better and better with more rectangles
Take the limits to infinity
Take the area calculation we did before, but with infinitely many \(ks\)
\[ \large \lim_{k\to\infty}R(f(x),\Delta x)=\text{True Area} \]
We formally call this an integral with the following notation:
\[ \large \int^b_af(x)dx \]
Verbally:
The Integral of \(f(x)\) from \(a\) to \(b\) with respect to \(x\)
How do we calculate
Fundamental Theorem of Calculus
If \(f(x)\) is continuous over an interval \([a,b]\), and the function \(F(x)\) is defined by:
\[ \large F(x)=\int^x_af(t)dt \]
then \(F'(x)=f(x)\) over \([a,b]\)
Fundamental Theorem of Calculus means that Integrals are just Anti-Derivatives
If we integrate a derivative, we should get the same function back as the original vice-versa
\[ \begin{align} \frac{d}{dx}\left[\int f(x)dx\right]&=f(x) &\text{Differentiation is the inverse of integration} \end{align} \]
\[ \begin{align} \int f'(x)&=f(x)+C &\text{Integration is the inverse of differentiation} \end{align} \]
Where the heck did that C come from?
Start with \(x^2-4x+1\)
\(\frac{d}{dx}=2x-4\)
If we integrate, we’ll have \(x^2-4x+C\)
There is no way of knowing what the x-intercept should be without additional information
We have to solve with an initial value problem
Why is Integration Useful?
Solve Differential Equations
i.e \(\frac{dy}{dx}=2x-4\)
Logistic Growth is one example
Total area of curves
Integration is tough
Before integrating ask yourself:
Follow many of the same rules for derivatives
Sum and Difference Rules
\[ \begin{align} \int[f(x)+g(x)]dx=\int f(x)dx+\int g(x)dx \\ \int[f(x)-g(x)]dx=\int f(x)dx-\int g(x)dx \end{align} \]
Rules of Integration
Constant Rules
\[ \begin{align} \int k dx&=kx+C \\ \int kf(x)dx&=k\int f(x)dx \end{align} \]
Rules of Integration
Power Rule
\[ \int x^ndx=\frac{x^{n+1}}{n+1}+C \text{, n}\ne1 \]
Examples
\[ \begin{align} f(x)&=4 & &g(x)=2x^2 &f(x)=x^3-4x \end{align} \]
Initial Value Problems
To find the C terms, we need extra information
Often times that comes from being given an initial value
The start of the period we know that \(y(0)=a\) or \(y(12)=b\)
We use this information to solve for what C should be
Initial Value Problem: Example
The marginal cost of producing \(x\) units of a product is:
\[ \frac{dC}{dx}=25-0.02x \]
Where C is the cost ($), and x is the number of units produced. Given that producing 2 units of product costs $10, what is the complete cost equation?
\[ \begin{align} C(x)=&25x-\frac{.02x^2}{2}+D & \text{Solve with Power Rule}\\ 10&=25(2)-\frac{0.02(2)^2}{2}+D &\text{ Sub in x=2}\\ -39.96&=D\\ C(x)&=25x-\frac{0.2x^2}{2}-39.96 \end{align} \]
Definite vs indefinite integrals
Definite integrals have start and end values (aka bounds)
We must evaluate between the intervals by
\[ \begin{align} \int^b_axdx\\ \frac{1}{2}x^2|^b_a\\ \frac{1}{2}(b)^2-\frac{1}{2}(a)^2 \end{align} \]
Indefinite Integrals no bounds
Need initial values to find integration constants
\[ \int f(x)dx \]
Definite Integral: Example
\[ \begin{align} \int^2_{-1}3x^2dx \\ x^3|^2_{-1}\\ (2)^3-(-1)^3=9 \end{align} \]
Area between curves
Area between curves example
Find the area of the shaded region in the graph below
Area between curves example
\[ \small \begin{align} \int^1_0 \sqrt{x}=\frac{2}{3}x^\frac{3}{2} \\ \int^1_0 x^2=\frac{1}{3}x^3 \end{align} \]
\[ \small \begin{align} \frac{2}{3}(1)^{\frac{3}{2}}-\frac{2}{3}(0)^{\frac{3}{2}}=\frac{2}{3}\\ \frac{1}{3}(1)^3-\frac{1}{3}(0)^3=\frac{1}{3} \end{align} \]
\[ \small \frac{2}{3}-\frac{1}{3}=\frac{1}{3} \]
Explain and demonstrate how to approximate the area under the curve \(f(x)=-9x^3+3x-9\) on the interval [4,10] using a Riemann sum with \(k=3\)
Take the Integrals
\[ \begin{align} \text{A) }y=\frac{3}{x^2}, y(0)=5& &\text{B) }g(t)=3t^5-2t^3+16t-7 & &\text{C) } \int^4_2\frac{1}{2}x \end{align} \]